Harer-Zagier formulas for knot matrix models
aa r X i v : . [ h e p - t h ] F e b Harer-Zagier formulas for knot matrix models
A. Morozov a,b,c, ∗ A. Popolitov a,b,c, † Sh. Shakirov b, ‡ ITEP/TH-03/21IITP/TH-03/21MIPT/TH-03/21 a Institute for Theoretical and Experimental Physics, Moscow 117218, Russia b Institute for Information Transmission Problems, Moscow 127994, Russia c Moscow Institute of Physics and Technology, Dolgoprudny 141701, Russia ∗ [email protected] † [email protected] ‡ [email protected] Abstract
Knot matrix models are defined so that the averages of characters are equal to knot polynomials. Fromthis definition one can extract single trace averages and generation functions for them in the group rank– which generalize the celebrated Harer-Zagier formulas for Hermitian matrix model. We describe theoutcome of this program for HOMFLY-PT polynomials of various knots. In particular, we claim that theHarer-Zagier formulas for torus knots factorize nicely, but this does not happen for other knots. This fact ismysteriously parallel to existence of explicit β = 1 eigenvalue model construction for torus knots only, andcan be responsible for problems with construction of a similar model for other knots. Superintegrability property usually means that some complete set of averages is explicitly calculable. Thefamous examples of this kind begin with harmonic oscillator and the motion in Coulomb potential. Eigenvaluematrix models [1–3] seem to possess this property in the following sense [4–8]: averages of characters are explicitlyknown – they are again characters [9, 10]. Moreover, the dependence on the size of the matrix N is captured inthe topological locus – the specialization of time-variables at the r.h.s. of the relation < character > ∼ character .Therefore the N -dependence is polynomial in q N for q -deformed models and just polynomial in N in the limit q −→
1. Laplace transform in N converts this average into a rational function.Remarkably, sometimes there is even more: not only denominator, but numerator is drastically simplified – allthe roots are just plus-or-minus powers of q . However, this happens for single-trace averages < P k > rather thancharacters and one needs to derive a special formula to describe the answer. We call these expressions the Harer-Zagier formulas (HZF), because they were first discovered in [11] for the simplest Gaussian Hermitian model(see [12, 13] for further developments in that case, including relation to other interesting subjects like Brezin-Hikami formulas [14], Okounkov’s exponentials [15] and W -representations [16]). Reversing the statement, theclaim is thatHZF for single-trace averages are rational functions with zeroes and poles at plus-or-minus powers of q (1)This is a frequent property of eigenvalue matrix models and it can serve as an alternative manifestation ofsuperintegrability phenomenon. An important question is what is the relation between the two manifestations,and, if different, which is more restrictive.In this paper we consider the first example when the difference occurs – the hypothetical knot matrixmodels, where superintegrability is assumed to imply that averages of characters are the corresponding coloredHOMFLY-PT polynomials. What we demonstrate is that this implies factorization of the single-trace averagesin the spirit of (1) only for torus knots and only for the Sl N – which so far remain the only case, when theeigenvalue formula (the β = 1 TBEM model [17, 18]) is actually known. Put differently, the statement is thatthe β = 1 TBEM matrix model for torus knots does possess the property (1) , while (1) does not follow fromassumption < character > = HOM F LY − P T (2)for non-torus knots. In fact our result questions the relevance of the postulate (2) beyond the torus-knot variety,i.e. implies that the problem of knot matrix models remains open. Knot matrix models
Matrix model for a knot K can be defined by the hypothetical/desirable superinterability property [7, 8, 19] h χ R i K := P K R ( q, A = q N ) (3)This definition is supported by existence of explicit TBEM model [17, 18] for torus knots, where the measure atthe l.h.s. contains peculiar deformations of triginometric Vanderomonde functions (see sec.3 below). Equallyexplicit formulas are not yet available even for twisted knots (see [20] for explanation of related beauties anddifficulties), but eq.(3) allows to bypass them. In fact, definition/ansatz (3) is very restrictive and allows for anumber of non-trivial consistency checks – a possibility, which we begin to explore in the present paper.On the other hand, ansatz (3) has some freedom. The characters χ R can be chosen as various symmetricpolynomials (Schur, Jack, Macdonald etc.) and P K R can be chosen to be various knot polynomials (HOMFLY-PT, Khovanov-Rozansky, super- or hyper-polynomials etc.) in a suitably chosen framing .All these details are crucial to the transition (3) → (1), and so current lack of explicit eigenvalue modelrepresentation for knot matrix models beyond torus knots can be, at least partly, attributed to difficulty offixing these freedoms simultaneously in exactly the right way.In cases when characters χ R are Schur functions (i.e. for β = 1 eigenvalue models), the single-trace operators(times) are expressed entirely through single-hook characters: P k = k − X i =0 ( − ) i Schur [ k − i, i ] { P } (4)while the simplest possibility for knot polynomials [21–28] are HOMFLY-PT polynomials. This implies that h P k i K N = k − X i =0 ( − ) i H K [ k − i, i ] ( q, q N ) (5)One can further perform a Laplace transform in N to get a (1-point) Harer-Zagier function Z K k := ∞ X N =0 λ N · h P k i K N (6)Also various generation functions w.r.t. the k -variables can be introduced, but while they lead to simplificationat q = 1, they seem to blur matters in the q -deformed case [29]. TBEM model [17, 18] provides explicit measure in (3) for torus knot K = Torus m,n and for the gauge group SU ( N ) (HOMFLY-PT invariants [21–28]): h F i Torus m,n ∼ I unitcircle F { e x i } N Y i 3) the fundamental HOMFLY-PT is H Torus , [1] = A { A }{ q } (cid:0) − A − { Aq }{ A/q } (cid:1) = ( q + q − ) q N − ( q + 1 + q − ) q N + q N q − q − (14)and we get a Harer-Zagier formula Z Torus , = 1 q − q − (cid:18) q + q − − q λ − q + 1 + q − − q λ + 11 − qλ (cid:19) = λq ( q − λ )(1 − qλ )(1 − q λ )(1 − q λ ) (15)which is a nicely factorized expression. It begins from λ , because the contribution of N = 0 (Alexanderpolynomial) is nullified by the factor D R .This answer can be easily generalized to other torus knots and other single-trace averages, for instance, for2-strand torus knots one has Z Torus ,n = λq n ( q n − λ )(1 − q n − λ )(1 − q n λ )(1 − q n +2 λ ) = λq n · ( q − n λ ; q ) ∞ ( q − n +2 λ ; q ) ∞ · ( q n +4 λ ; q ) ∞ ( q n − λ ; q ) ∞ (16)with the standard notation for the q -Pochhammer symbol( a ; q ) ∞ = ∞ Y k =0 (1 − q k a ) (17)3 Failures beyond torus fundamental HOMFLY In this section we list the cases, when factorization does not occur already in the simplest HZF – for theknot polynomial in the fundamental representation. What unifies these case – for torus and non-torus knots –is the lack of explicit eigenvalue matrix model, which converts characters into knot polynomials. If we define averages with the help of reduced polynomials, things would be different. In particular at N = 0we get at the r.h.s. the Alexander polynomials, which depend on representation only through the size of thesingle-hook diagram [31] h P k i K N =0 = k X i =0 ( − ) i A K [ k − i, i ] ( q ) = A [1] ( q k ) · k X i =0 ( − ) i ! = A K [1] ( q k ) · δ k, odd (18)From [30] for torus knots A Torus m,n [1] = ( q − q − )( q mn − q − mn )( q m − q − m )( q n − q − n ) (19)so h P k i K N =0 does have the structure (1), but this does not generalize to N = 0.Namely, the needed average would be h Schur R i Torus m,n SL ( N ) = X Q ∈ m | R | q n κ Qm · C QR · Schur Q n [ Nk ][ k ] o Schur R n [ Nk ][ k ] o (20)and the corresponding Harer-Zagier formula does not factorize already for trefoil Z Torus , = − q + q − − q λ − − q λ = q − q − − λq (1 − q λ )(1 − q λ ) (21)Factorization, however, occurs for the q -derivative Z Torus , = Z Torus , ( λq ) − Z Torus , ( λ/q ) q − q − (22)but there is no clear reason for it, besides pure technical on one hand and explicit existence of the TBEMeigenvalue model on the other hand. It is easy to check that at least some other knots in the fundamental representation do not possess such HZfactorization property. For example, for the figure-eight knot 4 the Harer-Zagier functions (for both normalizedand non-normalized HOMFLY-PT) do not factorize: H [1] = (cid:16) A + A − q − q + 1 (cid:17) (23) Z = − λ q + λq − λq − λq + λ − q +3 q − q ( λ − q ( λ − q )( λq − Z = Z ( λq ) − Z ( λ/q ) q − q − = λ ( λ q + λq − λq − λq − λq − λq + λ + q ) q ( λ − q )( λq − λ − q )( λq − (24) Likewise, the torus superpolynomials are also not suitable: P (2 , ∼ A ( A t + A q − − A − A − t − t − A = t N = ⇒ (25) Z (2 , = 1 t − t − (cid:18) t + q − − λt − t + 1 + q − − λt + 11 − λt (cid:19) = t q λ (cid:0) t q + t − q − λt (cid:1) (1 − λt )(1 − λt )(1 − λt ) t = q −→ λq ( q − λ )(1 − qλ )(1 − q λ )(1 − q λ ) Perhaps, for things to work, one needs to modify ( t -deform) the prescription (12) on how to do the Laplacetransform. However, at this point this is pure speculation, and this line of development will be pursued elsewhere.4 .4 Kauffman polynomials Kauffman polynomials are the analogues of HOMFLY-PT for the SO ( N ) gauge groups. They are alsodesribed by an analog of Rosso-Jones formula K Torus n,m R = q mn ( N − κ R q mn | λ | X | ν |≤ n | R | b νR,n q − mn ( N − κ ν q − mn | ν | d ν (26)where d ν is the quantum dimension of SO ( N ) representation, corresponding to diagram ν and coefficients b νR,n are the direct analog of Adams coefficients. The details of their calculation can be found in [32].In topological framing unreduced fundamental Kauffman satisfies K K (cid:3) , red − A − q )( Aq + 1)( A + 1)( A − 1) (27)From (26) one can deduce Kauffman polynomial for trefoil in fundamental representation K Torus , (cid:3) , red = − A · A ( q − q ) + A ( q − q + 1) − A ( q − q ) − ( q + 1) q (28)but its Laplace transform does not factorize. Moreover, this time it is not cured by multiplication with anyreasonable functions of A , even different from dimension. One can wonder what this means from the pointof view of our matrix model – factorization hypothesis. The answer is that the SO ( N ) models are assocaitedwith β = 2 rather than β = 1 (rouhgly speaking, 2 β is the power of Vandermonde-like factor in the measure).TBEM formula with β = 1 is easily generalized to simply-laced groups, which includes D , but not B – while(28) for Kauffman invariant unifies even and odd N through A = q N − – and the unifying matrix model shooulgave β = 2. This puts Kauffmann example into intermediate position, and once again calls for the properunderstanding in group theoretic terms of the Laplace transform, and its proper generalization.To summarize, we illustrated the distinguished role of HOMFLY-PT polynomials in torus family from thepoint of view of factorizability of HZF: any deviation seems to violate it. Instead, as we claim in the nextsection, for torus HOMLY-PT factorization is true for all single-trace HZF h p k i , not only for h p i . For higher representations of SU ( N ) the issue of framing becomes important, because we get a linear combi-nation of different representation in R ⊗ m , which transform differently under a change of framing. Factorizationtakes place in the so-called spectral (or vertical ) framing, which was used by Rosso and Jones in their originalformula for colored HOMFLY [33] and is exactly the framing, assumed in the Tierz-Brini-Eynard-Marino matrixmodel [17, 18], and in which the Ooguri-Vafa partition function is nicely expressed in terms of the free-fermionformalism [34, 35].Namely, the Rosso-Jones formula in this framing reads H Torus m,n R ( A, q ) = A n | R | X Q ∈ m | R | q n κ Qm · C QR · Schur Q (cid:26) p k = A k − A − k q k − q − k (cid:27) , (29)where C QR are, again, Adams coefficients from (9). Note that vertical framing explicitly breaks the n ↔ m symmetry.With this convention, the Laplace transform of single-trace correlator becomes nicely factorized: Z Torus n,m d = λq d nm · Q dm − i =0 (cid:16) − q d ( − n − m )+2+2 i λ (cid:17)Q dmi =0 (cid:16) − q d ( n − m )+2 i (cid:17) = λq d nm · ( λq d ( − n − m )+2 ; q ) ∞ ( λq d ( − n + m ) ; q ) ∞ ( λq d ( n + m )+2 ; q ) ∞ ( λq d ( n − m ) ; q ) ∞ where we once again use the q -Pochhammer symbols (17).This factorization formula for the Laplace transform in N of the peculiar combination of torus HOMPLY-PTpolynomials is the main result of this paper. 5he proof of this identity is calculational and combinatorial, very similar in spirit to that of [29]. The mainpoint is seen already at q = 1: according to (9), in this case we deal just with the dimension of representationSchur R { N } , whose Laplace transform is factorizable only for single-hook R , while factorization is lost beyondone-hook (and thus for multi-trace averages), e.g. P N λ N · Schur [3 , { N } = λ ( λ +3 λ +1)(1 − λ ) . Moreover, the single-hook factorization is preserved by the q -deformed Adams rule (9) and the action of ˆ W operator [16], which isresponsible for the factor q − n κ Qm in (29). This is technically straightforward: Z Torus n , m d = ∞ X N =0 λ N d − X i =0 ( − i H Torus n , m [ d − i, i ] ( q, q N )= ∞ X N =0 λ N q Ndn X Q ⊢ dm q nm κ Q d − X i =0 ( − i C Q [ d − i, i ] !| {z } dm − P L =0 ( − L δ Q, [ dm − L, L ] Schur Q (cid:18) p k = q Nk − q − Nk q k − q − k (cid:19) = ∞ X N =0 λ N q Ndn dm − X L =0 ( − L q nm κ [ dm − L, L ] Schur [ dm − L, L ] (cid:18) p k = q Nk − q − Nk q k − q − k (cid:19) = dm − X L =0 ( − L q nm dm ( dm − L − q dm − q − dm ∞ P N =0 λ N q Ndn dm − Q s =0 ( q N − L + s − q − N + L − s ) dm − L − Q s =1 ( q dm − L − s − q − dm + L + s ) L − Q s =0 ( q L − s − q − L + s )= dm − Q s =1 ( q s − q − s ) dm Q s =0 (1 − q d ( n − m )+2 s λ ) dm − X L =0 ( − L ( λq dn ) L +1 q dn ( dm − L − dm − L − Q s =1 ( q dm − L − s − q − dm + L + s ) L − Q s =0 ( q L − s − q − L + s )= λ q d nm dm − Q s =0 (1 − q d ( − n − m )+2+2 s λ ) dm Q s =0 (1 − q d ( n − m )+2 s λ )The main point here is the double application of the projection rule (4) to the definition (9) of Adams coefficients: X R ( − ) i · δ R, [ d − i, i ] · X Q ⊢ m | R | C QR · Schur R { p k } ( ) = Schur R { p km } = ⇒ X Q X i ( − ) i C Q [ d − i, i ] ! Schur Q { p k } = X i ( − ) i Schur [ d − i, i ] { p km } ( ) = p dm ( ) = X L ( − ) L · Schur [ dm − L, L ] { p k } = ⇒ d − X i =0 ( − ) i C Q [ d − i, i ] = dm − X L =0 ( − ) L δ q, [ dm − L, L ] (30)Despite apparent simplicity of this calculation, a conceptual proof is highly desirable, applicable to the wholevariety of β = 1 matrix models. In this paper we proposed to view the factorization of the Laplace transform of single-trace average as analternative manifestation of superintegrability .Remarkably, this factorization turns out to be present for torus knots’ HOMFLY-PT polynomials, where theeigenvalue model (and free-fermion representation) is explictly known, but the very naive attempts to observesimilar factorization for slight deformations of this setting: to superpolynomials, to other knots and even toKauffman polynomials – all fail. 6his seems to suggest, that the problem of finding proper matrix models for families of knot polynomialsis more tricky and rigid than was first thought. One of the ways around this situation would be to relax theprescription h character i = knot polynomial (31)in some yet unknown way. As was recently demonstrated [36] similar broadening of the point of view can bevery fruitful in discovering new character expansion formulas.Another possibility would be to understand the group-theoretic meaning of the Laplace transform, and toadjust it, accordingly. At the moment this part of the Harer-Zagier construction seems completely ad hoc .There is a number of possible directions/questions to pursue, which we would like to point out • What is the formula for the double-trace correlators? Is it, in some sense, similar to the one for q -deformedHermitian Gaussian matrix model? • HOMFLY-PT polynomials for torus knots have a well-known generalization from S to Seifert spaces.The knot matrix model is known for this case. Does the Harer-Zagier factorization persist as well? • Superpolynomials, which we considered in this paper, are not Khovanov-Rozansky polynomials. Instead,they coincide for large enough N (which is knot-dependent). Therefore, the Laplace transformed sumsfor superpolynomials and actual KR polynomials differ by some polynomial in λ . Can it be, that thispolynomial transforms non-factorizable Harer-Zagier function into factorizable? • Last but not least, there is a question about the relation of superintegrability (in the form of Harer-Zagierfactorization) to other well-known, and undergoing rapid development, knot-theoretical structures: theknots-quivers correspondence [37–40] and theory of q-Virasoro localization [41–43].We hope to address some, or all of these questions in future. Acknowledgements The work is partly supported by RFBR grants 19-02-00815 (A.M., Sh.Sh.), 19-51-18006 Bolg-a (A.M.),19-01-00680 (A.P.) and by the joint RFBR-MOST grant 21-52-52004 (A.M., A.P.) References [1] Morozov, A. “Matrix models as integrable systems”. In: “Particles and fields”, pp. 127–210. Springer(1999). arXiv:hep-th/9502091 [hep-th] .[2] Morozov, A. “Challenges of matrix models”. NATO Science Series II: Mathematics, Physics and Chemistry,p. 129–162 DOI:10.1007/1-4020-3733-3_6 (2005). arXiv:hep-th/0502010 [hep-th] .[3] Mironov, A. “Quantum deformations of τ -functions, bilinear identities and representation theory”. Symme-tries and Integrability of Difference Equations, vol. 9:pp. 219–2 (1996). arXiv:hep-th/9409190 [hep-th] .[4] Itoyama, H., Mironov, A. and Morozov, A. “Ward identities and combinatorics of rainbow tensor models”.Journal of High Energy Physics, vol. 2017(6) DOI:10.1007/jhep06(2017)115 (2017). arXiv:1704.08648 [hep-th] .[5] Itoyama, H., Mironov, A. and Morozov, A. “Tensorial generalization of characters”. Journal of High EnergyPhysics, vol. 2019(12) DOI:10.1007/jhep12(2019)127 (2019). arXiv:1909.06921 [hep-th] .[6] Itoyama, H., Mironov, A. and Morozov, A. “Complete solution to Gaussian tensor model and its integrableproperties”. Physics Letters B, vol. 802:p. 135237 DOI:10.1016/j.physletb.2020.135237 (2020). arXiv:1910.03261 [hep-th] .[7] Mironov, A. and Morozov, A. “On the complete perturbative solution of one-matrix models”. PhysicsLetters B, vol. 771:p. 503–507 DOI:10.1016/j.physletb.2017.05.094 (2017). arXiv:1705.00976 [hep-th] .78] Mironov, A. and Morozov, A. “Sum rules for characters from character-preservation property of matrixmodels”. Journal of High Energy Physics, vol. 2018(8) DOI:10.1007/jhep08(2018)163 (2018). arXiv:1807.02409 [hep-th] .[9] Itoyama, H., Mironov, A., Morozov, A. and Morozov, A. “Character expansion for HOMFLY polynomialsIII: All 3-strand braids in the first symmetric representation”. International Journal of Modern Physics A,vol. 27(19):p. 1250099 DOI:10.1142/s0217751x12500996 (2012). arXiv:1204.4785 [hep-th] .[10] Mironov, A., Morozov, A., Shakirov, S. and Smirnov, A. “Proving AGT conjecture as HS duality: Extensionto five dimensions”. Nuclear Physics B, vol. 855(1):p. 128–151 DOI:10.1016/j.nuclphysb.2011.09.021 (2012). arXiv:1105.0948 [hep-th] .[11] Harer, J. and Zagier, D. “The Euler characteristic of the moduli space of curves”. Inventiones mathematicae,vol. 85(3):pp. 457–485 (1986).[12] Morozov, A. and Shakirov, S. “Exact 2-point function in Hermitian matrix model”. Journal of High EnergyPhysics, vol. 2009(12):p. 003 (2009). arXiv:0906.0036 [hep-th] .[13] Morozov, A. and Shakirov, S. “From Brezin-Hikami to Harer-Zagier formulas for Gaussian correlators”(2010). arXiv:1007.4100 [hep-th] .[14] Br´ezin, E. and Hikami, S. “Duality and replicas for a unitary matrix model”. Journal of High EnergyPhysics, vol. 2010(7):p. 67 (2010). arXiv:1005.4730 [hep-th] .[15] Okounkov, A. “Generating functions for intersection numbers on moduli spaces of curves”. InternationalMathematics Research Notices, vol. 2002(18):p. 933 DOI:10.1155/s1073792802110099 (2002). arXiv:math/0101201 [math.AG] .[16] Morozov, A. and Shakirov, S. “Generation of matrix models by W-operators”. Journal of High EnergyPhysics, vol. 2009(04):p. 064 (2009). arXiv:0902.2627 [hep-th] .[17] Tierz, M. “Soft matrix models and Chern–Simons partition functions”. Modern Physics Letters A,vol. 19(18):p. 1365–1378 DOI:10.1142/s0217732304014100 (2004). arXiv:hep-th/0212128 [hep-th] .[18] Brini, A., Mari˜no, M. and Eynard, B. “Torus knots and mirror symmetry”. In: “Annales Henri Poincar´e”,vol. 13, pp. 1873–1910. Springer (2012). arXiv:1105.2012 [hep-th] .[19] Mironov, A. and Morozov, A. “Superintegrability and Kontsevich-Hermitian relation” (2021). arXiv:2102.01473 [hep-th] .[20] Alexandrov, A., Mironov, A., Morozov, A. and Morozov, A. “Towards matrix model representation ofHOMFLY polynomials”. JETP Letters, vol. 100(4):p. 271–278 DOI:10.1134/s0021364014160036 (2014). arXiv:1407.3754 [hep-th] .[21] Alexander, J. W. “Topological invariants of knots and links”. Transactions of the American MathematicalSociety, vol. 30(2):pp. 275–306 (1928).[22] Conway, J. H. “An enumeration of knots and links, and some of their algebraic properties”. In: “Compu-tational problems in abstract algebra”, pp. 329–358. Elsevier (1970).[23] Jones, V. F. “Index for subfactors”. Inventiones mathematicae, vol. 72(1):pp. 1–25 (1983).[24] Jones, V. F. “A polynomial invariant for knots via von Neumann algebras”. In: “Fields Medallists’Lectures”, pp. 448–458. World Scientific (1997).[25] Jones, V. F. “Hecke algebra representations of braid groups and link polynomials”. In: “New DevelopmentsIn The Theory Of Knots”, pp. 20–73. World Scientific (1987).[26] Kauffman, L. H. “State models and the Jones polynomial”. Topology, vol. 26(3):pp. 395–407 (1987).[27] Freyd, P., Yetter, D., Hoste, J., Lickorish, W. R., Millett, K. and Ocneanu, A. “A new polynomial invariantof knots and links”. Bulletin of the American Mathematical Society, vol. 12(2):pp. 239–246 (1985).828] Przytycki, J. and K.P., T. “Invariants of links of Conway type”. Kobe J. Math., vol. 4:p. 115–139 (1987). arXiv:1610.06679 [math.GT] .[29] Morozov, A., Popolitov, A. and Shakirov, S. “Quantization of Harer-Zagier formulas”. Physics Letters B,vol. 811:p. 135932 (2020). arXiv:2008.09577 [hep-th] .[30] Dunin-Barkowski, P., Mironov, A., Morozov, A., Sleptsov, A. and Smirnov, A. “Superpolynomials for torusknots from evolution induced by cut-and-join operators”. Journal of High Energy Physics, vol. 2013(3) DOI:10.1007/jhep03(2013)021 (2013). arXiv:1106.4305 [hep-th] .[31] Mironov, A. and Morozov, A. “Eigenvalue conjecture and colored Alexander polynomials”. The EuropeanPhysical Journal C, vol. 78(4) DOI:10.1140/epjc/s10052-018-5765-5 (2018). arXiv:1610.03043 [hep-th] .[32] Stevan, S. “Knot invariants, Chern–Simons theory and the topological recursion”. Ph.D. thesis, Universityof Geneva (2014). DOI:10.13097/archive-ouverte/unige:41515 [33] Rosso, M. and Jones, V. “J. knot theory ramifications” (1993).[34] Dunin-Barkowski, P., Popolitov, A., Shadrin, S. and Sleptsov, A. “Combinatorial structure of coloredHOMFLY-PT polynomials for torus knots”. Communications in Number Theory and Physics, vol. 13(4):p.763–826 DOI:10.4310/cntp.2019.v13.n4.a3 (2019). arXiv:1712.08614 [math-ph] .[35] Dunin-Barkowski, P., Kazarian, M., Popolitov, A., Shadrin, S. and Sleptsov, A. “Topological recursion forthe extended Ooguri-Vafa partition function of colored HOMFLY-PT polynomials of torus knots” (2020). arXiv:2010.11021 [math-ph] .[36] Mironov, A. and Morozov, A. “Superintegrability of Kontsevich matrix model” (2020). arXiv:2011.12917 [hep-th] .[37] Kucharski, P. “Quivers for 3-manifolds: the correspondence, BPS states, and 3d N = 2 theories”. Journalof High Energy Physics, vol. 2020(9) DOI:10.1007/jhep09(2020)075 (2020). arXiv:2005.13394 [hep-th] .[38] Ekholm, T., Gruen, A., Gukov, S., Kucharski, P., Park, S. and Su lkowski, P. “ b Z at large N: from curvecounts to quantum modularity” (2020). arXiv:2005.13349 [hep-th] .[39] Ekholm, T., Kucharski, P. and Longhi, P. “Multi-cover skeins, quivers, and 3d N = 2 dualities”. Journalof High Energy Physics, vol. 2020(2) DOI:10.1007/jhep02(2020)018 (2020). arXiv:1910.06193 [hep-th] .[40] Kucharski, P., Reineke, M., Stoˇsi´c, M. and Su lkowski, P. “Knots-quivers correspondence”. Advances inTheoretical and Mathematical Physics, vol. 23(7):p. 1849–1902 DOI:10.4310/atmp.2019.v23.n7.a4 (2019). arXiv:1707.04017 [hep-th] .[41] Cassia, L., Lodin, R. and Zabzine, M. “On matrix models and their q -deformations” (2020). arXiv:2007.10354 [hep-th] .[42] Nedelin, A., Nieri, F. and Zabzine, M. “q-Virasoro modular double and 3d partition functions”. Commu-nications in Mathematical Physics, vol. 353(3):p. 1059–1102 DOI:10.1007/s00220-017-2882-1 (2017). arXiv:1605.07029 [hep-th] .[43] Nedelin, A. and Zabzine, M. “q-Virasoro constraints in matrix models”. Journal of High Energy Physics,vol. 2017(3) DOI:10.1007/jhep03(2017)098 (2017). arXiv:1511.03471 [hep-th]arXiv:1511.03471 [hep-th]